Roz Chast. Ben Schwartz. David Sipress. They're the familiar names at the bottom of The New Yorker's cartoons, the witty ones whose work seems to make it into most editions. But as Morley Safer finds out, they get rejected about 95 percent of the time -- and the magazine likes their work! Safer reports on the cartoon selection process at The New Yorker, interviewing the cartoonists, their editor, Bob Mankoff, and The New Yorker editor David Remnick. Safer's story will be broadcast on 60 Minutes, Sunday, March 23 at 7 p.m. ET/PT.

The New Yorker allowed 60 Minutes cameras to record Mankoff's regular Wednesday meeting with cartoonists, where he selects the 17 drawings readers see in the magazine each week. He looks at hundreds. The veterans know all too well. "We all probably do...700 or 800 cartoons a year we hand in...We're lucky if we sell 30 cartoons a year," says Sipress. And when you think you hit it out of the park? "That, right away, that goes in the garbage," Chast tells Safer. Newcomer Charlie Hankin says at first he was addicted to the rejection. "It makes you feel alive."

"I know what it feels like. It feels a little bit like a punch in the stomach," says Mankoff, who estimates he submitted about 2,000 pieces before getting accepted and eventually becoming The New Yorker cartoon editor. Mankoff earned his editorship with the right cartoons, including the classic of the businessman on the phone captioned: "Thursday's out. How about Never? Is never good for you?"

Once Mankoff has done his job, it's up to Remnick to make the final decision. As Remnick makes his choices, Safer says he does not get one particular cartoon Remnick has picked. Remnick says he is not alone. "At least five times a week somebody will come up to me and say, 'I didn't get such and such a cartoon,'" he tells Safer. "Here is the deep secret: including me once in a while. I will pick a raft of cartoons. And then later it'll come time to run this cartoon and I'll look at it, and I won't quite get it anymore. Because sometimes the grenade goes off in the moment and then it doesn't repeat down the line," says Remnick.

Told by Safer that some believe there's a cartoon in each edition of The New Yorker that no one is meant to understand," a laughing Remnick responds. "I'm going to keep that myth alive."